A153653: Triangle T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2j + 1)prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 8, read by rows.

A153653 ↗ palindromicγ-positivenonnegative coefficients

Proved for all n all roots real, in (−∞, 0)coefficients of one sign Hadamard kernel interlacing interlacing (weak)

Recurrence from A153653

\(\left(-2 + n\right)\,P_{n}(x) = \left(-2 - 2\,x + n + n x\right)\,P_{n-1}(x) + \left(-646\,x + 323\,n x\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 19 + 19\,x\)

\(P_{2}(x) = 2 + 718\,x + 2\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	19	19														38
2	2	718	2													722
3	2	6857	6857	2												13718
4	2	7505	245628	7505	2											260642
5	2	8153	2467944	2467944	8153	2										4952198
6	2	8801	4900212	84273732	4900212	8801	2									94091762
7	2	9449	7542432	886319856	886319856	7542432	9449	2								1787743478
8	2	10097	10394604	2476630764	28993055148	2476630764	10394604	10097	2							33967126082
9	2	10745	13456728	4923230904	317750999400	317750999400	4923230904	13456728	10745	2						645375395558
10	2	11393	16728804	8294144724	1122625967076	10000258811604	1122625967076	8294144724	16728804	11393	2					12262132515602
11	2	12041	20210832	12657396672	2721123693792	113756457584880	113756457584880	2721123693792	12657396672	20210832	12041	2				232980517796438
12	2	12689	23902812	18081011196	5412789836316	479085768644220	3457596511317852	479085768644220	5412789836316	18081011196	23902812	12689	2			4426629838132322
13	2	13337	27804744	24633012744	9519209972568	1363421511575352	40680018079878312	40680018079878312	1363421511575352	9519209972568	24633012744	27804744	13337	2		84105966924514118
14	2	13985	31916628	32381425764	15384009601620	3121271838677988	196788142863536724	1198163709315422820	196788142863536724	3121271838677988	15384009601620	32381425764	31916628	13985	2	1598013371565768242

b-files: A153653, A153653 (reversed triangle) ?

Technical details

Node id: n-71e25d00553f2c01 · revision 0 · JSON
Definition source: oeis_catalog:A153653
Fingerprint: f6-fac9c3c65d64b266

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: yes
γ-positive (gamma_positive): yes
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

A153653 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?
A153653 reversed triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  2;
  1: 19,    19;
  2:  2,   718,        2;
  3:  2,  6857,     6857,          2;
  4:  2,  7505,   245628,       7505,            2;
  5:  2,  8153,  2467944,    2467944,         8153,            2;
  6:  2,  8801,  4900212,   84273732,      4900212,         8801,          2;
  7:  2,  9449,  7542432,  886319856,    886319856,      7542432,       9449,        2;
  8:  2, 10097, 10394604, 2476630764,  28993055148,   2476630764,   10394604,    10097,     2;
  9:  2, 10745, 13456728, 4923230904, 317750999400, 317750999400, 4923230904, 13456728, 10745, 2;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(n-2)*x)*P(n-1,x) + (323*n-646)*x*P(n-2,x) for n >= 3, with P(0,x) = 2, P(1,x) = 19 + 19*x, P(2,x) = 2 + 718*x + 2*x^2.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 19 + 19*x;
P[2, x_] := 2 + 718*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n - 2)*x)*P[n - 1, x] + (323*n - 646)*x*P[n - 2, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 19 + 19*x; v[3] = 2 + 718*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1] + (323*n - 646)*x*v[i - 2])/(n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A153653 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1:  2;
   2: 19,    19;
   3:  2,   718,        2;
   4:  2,  6857,     6857,          2;
   5:  2,  7505,   245628,       7505,            2;
   6:  2,  8153,  2467944,    2467944,         8153,            2;
   7:  2,  8801,  4900212,   84273732,      4900212,         8801,          2;
   8:  2,  9449,  7542432,  886319856,    886319856,      7542432,       9449,        2;
   9:  2, 10097, 10394604, 2476630764,  28993055148,   2476630764,   10394604,    10097,     2;
  10:  2, 10745, 13456728, 4923230904, 317750999400, 317750999400, 4923230904, 13456728, 10745, 2;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (n-3+(n-3)*x)*P(n-1,x) + (323*n-969)*x*P(n-2,x) for n >= 4, with P(1,x) = 2, P(2,x) = 19 + 19*x, P(3,x) = 2 + 718*x + 2*x^2.

%t Mathematica

P[1, x_] := 2;
P[2, x_] := 19 + 19*x;
P[3, x_] := 2 + 718*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n - 3)*x)*P[n - 1, x] + (323*n - 969)*x*P[n - 2, x])/(n - 3)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 2; v[2] = 19 + 19*x; v[3] = 2 + 718*x + 2*x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n - 3)*x)*v[i - 1] + (323*n - 969)*x*v[i - 2])/(n - 3)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A153653 reversed triangle, offset 1, from P₀

%e Example

Triangle begins:
   1:  2;
   2: 19,    19;
   3:  2,   718,        2;
   4:  2,  6857,     6857,          2;
   5:  2,  7505,   245628,       7505,            2;
   6:  2,  8153,  2467944,    2467944,         8153,            2;
   7:  2,  8801,  4900212,   84273732,      4900212,         8801,          2;
   8:  2,  9449,  7542432,  886319856,    886319856,      7542432,       9449,        2;
   9:  2, 10097, 10394604, 2476630764,  28993055148,   2476630764,   10394604,    10097,     2;
  10:  2, 10745, 13456728, 4923230904, 317750999400, 317750999400, 4923230904, 13456728, 10745, 2;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n-3)*P(n,x) = (n-3+(n-3)*x)*P(n-1,x) + (323*n-969)*x*P(n-2,x) for n >= 4, with P(1,x) = 2, P(2,x) = 19 + 19*x, P(3,x) = 2 + 718*x + 2*x^2.

%t Mathematica

P[1, x_] := 2;
P[2, x_] := 19 + 19*x;
P[3, x_] := 2 + 718*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n - 3)*x)*P[n - 1, x] + (323*n - 969)*x*P[n - 2, x])/(n - 3)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 2; v[2] = 19 + 19*x; v[3] = 2 + 718*x + 2*x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n - 3)*x)*v[i - 1] + (323*n - 969)*x*v[i - 2])/(n - 3)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
by Liu–Wang theorem with a lag row, Liu–Wang theorem for three-term recurrences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem with a lag row, Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem with a lag row, Liu–Wang theorem for three-term recurrences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m ≥ 2
by Liu–Wang theorem with a lag row
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
by Liu–Wang theorem for three-term recurrences

Proof chains ? JSON

Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003